Stable geometric dimension of vector bundles over even-dimensional real projective spaces

Bendersky, Martin; Davis, Donald; Mahowald, Mark

doi:10.1090/S0002-9947-05-03736-0

Stable geometric dimension of vector bundles over even-dimensional real projective spaces
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by Martin Bendersky, Donald M. Davis and Mark Mahowald PDF

Trans. Amer. Math. Soc. 358 (2006), 1585-1603 Request permission

Abstract:

In 1981, Davis, Gitler, and Mahowald determined the geometric dimension of stable vector bundles of order $2^e$ over $RP^{n}$ if $n$ is even and sufficiently large and $e\ge 75$. In this paper, we use the Bendersky-Davis computation of $v_1^{-1}\pi _*(SO(m))$ to show that the 1981 result extends to all $e\ge 5$ (still provided that $n$ is sufficiently large). If $e\le 4$, the result is often different due to anomalies in the formula for $v_1^{-1}\pi _*(SO(m))$ when $m\le 8$, but we also determine the stable geometric dimension in these cases.

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